Klein-Gordon equation

• free massive scalar field
• in condensed matter physics it describes long wavelength optical phonons

 

 

• relativistic generalization of Schrodinger equation
• this describes the spin-0 particles
• there are real KG equation and complex KG equation
  
  • real case describes electrically neutral particles
  • complex case describes charged particles
• \((\Box + m^2) \psi = 0\)
• people found 2 problems of KG equations
  
  • negative energy states
  • negative probability density
• correct interpretations of \(\phi\) requires the idea of quantum field rather than the particle wavefunction
  
  • negative probability density -> charge density
• Dirac suggested Dirac sea by invoking the exclusion principle and then KG equation only applicable to spinless particles
  
  • for example, \(\pi\)-meson
• Thus the Dirac equation comes in to deal with spin-\(1/2\) particles.